How generally can an AdS/CFT analogy be made?

[atyy]

Can every QFT/CFT be interpreted at least approximately as a theory of gravity? If not, is it known which QFT/CFTs have gravitational counterparts?

 

[Physics Monkey]

There are some of us who think so, but probably only with a very liberal usage of the term "gravity". Per usual AdS/CFT lore, any CFT/QFT with a stress tensor will correspond to something with a fluctuating metric in the bulk. However, there may be many more massless fields in the bulk, for example, an infinite tower in the case of proposed duals for the ordinary $$O(n)$$ model. The gravity theory may also be extremely far from its semiclassical regime. For example, the $$\mathcal{N} = 4$$ theory at very small coupling and with few colors is probably still dual to a string theory, but I think it would be very hard to talk meaningfully about an emergent dimension in any semiclassical sense.

So all the various inclusions you might ask about are unknown. We don't know if all holographic dualities complete to string theory on the gravity side. We don't know if all QFTs are holographic. We don't know if gravity can be included on the QFT side of holographic setup.

However, there is reason to be hopeful that the statement is very general. Even something as simple as the matrix harmonic oscillator is holographic with a (strange) string theory dual!

 

[atyy]

We don't know if gravity can be included on the QFT side of holographic setup.

Are there any proposals like that? Or where the QFT side is on a curved spacetime?

 

[Physics Monkey]

It's relatively easy to put the QFT on a fixed curved spacetime. For example, the $$\mathcal{N} = 4$$ theory on a spatial 3-sphere is dual to string theory in asymptotically "global" AdS. One can also do more complicated curved surfaces. The flat space limit (infinite sphere radius) in the QFT corresponds to a certain section of AdS called the Poincare patch.

Putting dynamical gravity in the boundary theory is much less certain. Randall-Sundrum models achieve this in a certain sense, because there are normalizable "graviton" modes on the brane. But it's not exactly the same thing as holographic duality. There are proposals that holography in de Sitter may require gravity in the dual QFT. For example, there may again be a normalizable "graviton" mode at the "boundary". In the case of de Sitter, one of the proposed duals is in two lower dimensions and with gravity, so that both "time" (a la Wheeler-deWitt/Hamiltonian constraint) and the "radial" coordinate emerge.

 

[arivero]

Hmm. What do you get if you put a 1+1 CFT in the QFT side?

 

[Physics Monkey]

Hmm. What do you get if you put a 1+1 CFT in the QFT side?

You get AdS_3. The CFT at finite temperature is dual to the BTZ black hole in AdS_3.

 

[arivero]

You get AdS_3. The CFT at finite temperature is dual to the BTZ black hole in AdS_3.

So String theory is a limit of AdS_3 by itself? Or this CFT can not be used to build string theories in the worldsurface?

 

[Physics Monkey]

The CFTs described via holographic duality by AdS_3 do not contain two dimensional gravity, but conformal field theories on the string worldsheet satisfy additional constraints related to 2d gravity and Weyl invariance.

For example, keeping the Weyl gauge redundancy intact on the string worldsheet requires the conformal field theory to have zero central charge. This necessitates the introduction of ghosts with negative central charge to cancel the positive central charge of other world sheet fields. BRST symmetry on the worldsheet protects you from anything nasty happening, but the resulting theory isn't like a run of the mill CFT. This is one way to determine the critical dimension of the particular flat vacuum considered.

CFTs described by AdS_3 have positive central charge related to the cosmological constant and the Newton constant by the famous formula of Brown and Henneaux. In fact, the best understood cases have a very large central charge. Haha, so this would make these CFTs sort of maximally unfriendly on the string world sheet.

PS: As an interesting side note, there are suggestions that CFTs potentially relevant for de Sitter may contain 2d gravity and hence would be more like theories on the string world sheet.